{"id":10837,"date":"2017-06-05T21:25:23","date_gmt":"2017-06-05T21:25:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10837"},"modified":"2020-10-22T09:28:51","modified_gmt":"2020-10-22T09:28:51","slug":"multiplying-two-binomials-using-the-distributive-property","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/multiplying-two-binomials-using-the-distributive-property\/","title":{"raw":"12.2.d - Multiplying Binomials","rendered":"12.2.d &#8211; Multiplying Binomials"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Multiply binomials<\/li>\r\n<\/ul>\r\n<\/div>\r\nJust like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial.\r\n<h3>Using the Distributive Property<\/h3>\r\nWe will start by using the Distributive Property. Look again at the following example.\r\n<table id=\"eip-id1168466233256\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x plus 3, times p, with red arrows from the p to the x and to the 3. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224428\/CNX_BMath_Figure_10_03_049_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We distributed the [latex]p[\/latex] to get<\/td>\r\n<td>[latex]x\\color{red}{p}+3\\color{red}{p}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>What if we have [latex]\\left(x+7\\right)[\/latex] instead of [latex]p[\/latex] ?\r\n\r\nThink of the [latex]x+7[\/latex] as the [latex]\\color{red}{p}[\/latex] above.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224430\/CNX_BMath_Figure_10_03_049_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute [latex]\\left(x+7\\right)[\/latex] .<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224432\/CNX_BMath_Figure_10_03_049_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute again.<\/td>\r\n<td>[latex]{x}^{2}+7x+3x+21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]{x}^{2}+10x+21[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that before combining like terms, we had four terms. We multiplied the two terms of the first binomial by the two terms of the second binomial\u2014four multiplications.\r\n\r\nBe careful to distinguish between a sum and a product.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{cccc}\\hfill \\mathbf{\\text{Sum}}\\hfill &amp; &amp; &amp; \\hfill \\mathbf{\\text{Product}}\\hfill \\\\ \\hfill x+x\\hfill &amp; &amp; &amp; \\hfill x\\cdot x\\hfill \\\\ \\hfill 2x\\hfill &amp; &amp; &amp; \\hfill {x}^{2}\\hfill \\\\ \\hfill \\text{combine like terms}\\hfill &amp; &amp; &amp; \\hfill \\text{add exponents of like bases}\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(x+6\\right)\\left(x+8\\right)[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168468281692\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x plus 6 times parentheses x plus 8. The next line shows parentheses x plus 6 times red parentheses x plus 8, with red arrows from x plus 8 to x and to 6. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(x+6\\right)\\left(x+8\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224433\/CNX_BMath_Figure_10_03_050_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute [latex]\\left(x+8\\right)[\/latex] .<\/td>\r\n<td>[latex]x\\color{red}{(x+8)}+6\\color{red}{(x+8)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute again.<\/td>\r\n<td>[latex]{x}^{2}+8x+6x+48[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{x}^{2}+14x+48[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146207[\/ohm_question]\r\n\r\n<\/div>\r\nNow we'll see how to multiply binomials where the variable has a coefficient.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(2x+9\\right)\\left(3x+4\\right)[\/latex]\r\n[reveal-answer q=\"901421\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"901421\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467332174\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 2x plus 9 times parentheses 3x plus 4. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(2x+9\\right)\\left(3x+4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute. [latex]\\left(3x+4\\right)[\/latex]<\/td>\r\n<td>[latex]2x\\color{red}{(3x+4)}+9\\color{red}{(3x+4)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute again.<\/td>\r\n<td>[latex]6{x}^{2}+8x+27x+36[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]6{x}^{2}+35x+36[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146208[\/ohm_question]\r\n\r\n<\/div>\r\nIn the previous examples, the binomials were sums. When there are differences, we pay special attention to make sure the signs of the product are correct.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(4y+3\\right)\\left(6y - 5\\right)[\/latex]\r\n[reveal-answer q=\"834420\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"834420\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469625351\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 4y plus 3 times parentheses 6y minus 5. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(4y+3\\right)\\left(6y - 5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]4y\\color{red}{(6y-5)}+3\\color{red}{(6y-5)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute again.<\/td>\r\n<td>[latex]24{y}^{2}-20y+18y - 15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]24{y}^{2}-2y - 15[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146209[\/ohm_question]\r\n\r\n<\/div>\r\nUp to this point, the product of two binomials has been a trinomial. This is not always the case.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(x+2\\right)\\left(x-y\\right)[\/latex]\r\n[reveal-answer q=\"982155\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"982155\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468254725\" class=\"unnumbered unstyled\" summary=\"The top line says parentheses x plus 2 times parentheses x minus y. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex](x+2)(x-y)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]x\\color{red}{(x-y)}+2\\color{red}{(x-y)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute again.<\/td>\r\n<td>[latex]x^2-xy+2x-2y[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>There are no like terms to combine.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146210[\/ohm_question]\r\n\r\n<\/div>\r\n<h3>Area Model for Multiplying Binomials<\/h3>\r\nNow let's explore multiplying two binomials. For those of you that use pictures to learn, you can draw an area model to help make sense of the process. You'll use each binomial as one of the dimensions of a rectangle, and their product as the area.\r\n\r\nThe model below shows [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]:\r\n\r\n[caption id=\"attachment_4607\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-4607\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/04191552\/Screen-Shot-2016-06-04-at-12.15.06-PM-300x290.png\" alt=\"Visual representation of multiplying two binomials.\" width=\"300\" height=\"290\" \/> Visual representation of multiplying two binomials.[\/caption]\r\n\r\nEach binomial is expanded into variable terms and constants, [latex]x+4[\/latex], along the top of the model and [latex]x+2[\/latex] along the left side. The product of each pair of terms is a colored rectangle. The total area is the sum of all of these small rectangles, [latex]x^{2}+2x+4x+8[\/latex], If you combine all the like terms, you can write the product, or area, as [latex]x^{2}+6x+8[\/latex].\r\n\r\nYou can use the distributive property to determine the product of two binomials.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]\r\n[reveal-answer q=\"186797\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"186797\"]Distribute the [latex]x[\/latex] over [latex]x+2[\/latex], then distribute 4 over [latex]x+2[\/latex].\r\n<p style=\"text-align: center\">[latex]x\\left(x\\right)+x\\left(2\\right)+4\\left(x\\right)+4\\left(2\\right)[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center\">[latex]x^{2}+2x+4x+8[\/latex]<\/p>\r\nCombine like terms [latex]\\left(2x+4x\\right)[\/latex].\r\n<p style=\"text-align: center\">[latex]x^{2}+6x+8[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(x+4\\right)\\left(2x+2\\right)=x^{2}+6x+8[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nLook back at the model above to see where each piece of [latex]x^{2}+2x+4x+8[\/latex] comes from. Can you see where you multiply [latex]x[\/latex] by [latex]x + 2[\/latex], and where you get [latex]x^{2}[\/latex]\u00a0from [latex]x\\left(x\\right)[\/latex]?\r\n\r\nAnother way to look at multiplying binomials is to see that each term in one binomial is multiplied by each term in the other binomial. Look at the example above: the [latex]x[\/latex] in [latex]x+4[\/latex] gets multiplied by both the [latex]x[\/latex] and the [latex]2[\/latex] from [latex]x+2[\/latex], and the [latex]4[\/latex] gets multiplied by both the [latex]x[\/latex] and the [latex]2[\/latex].\r\n\r\nThe following video provides an example of multiplying two binomials using an area model as well as repeated distribution.\r\n\r\nhttps:\/\/youtu.be\/u4Hgl0BrUlo\r\n<h2>The Table Method<\/h2>\r\nYou may see a binomial multiplied by itself written as\u00a0[latex]{\\left(x+3\\right)}^{2}[\/latex] instead of\u00a0[latex]\\left(x+3\\right)\\left(x+3\\right)[\/latex]. \u00a0To find this product, let's use another method. We will place the terms of each binomial along the top row and first column of a table, like this:\r\n<table style=\"width: 50%\">\r\n<thead>\r\n<tr>\r\n<td style=\"width: 23.2394%\"><\/td>\r\n<td style=\"width: 38.2629%\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 38.2629%\">[latex]+3[\/latex]<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 23.2394%\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 38.2629%\"><\/td>\r\n<td style=\"width: 38.2629%\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 23.2394%\">[latex]+3[\/latex]<\/td>\r\n<td style=\"width: 38.2629%\"><\/td>\r\n<td style=\"width: 38.2629%\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow multiply the term in each column by the term in each row to get the terms of the resulting polynomial. Note how we keep the signs on the terms, even when they are positive, this will help us write the new polynomial.\r\n<table style=\"width: 50%\">\r\n<thead>\r\n<tr>\r\n<td style=\"width: 23.2394%\"><\/td>\r\n<td style=\"width: 38.2629%\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 38.2629%\">[latex]+3[\/latex]<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 23.2394%\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 38.2629%\">[latex]x\\cdot{x}=x^2[\/latex]<\/td>\r\n<td style=\"width: 38.2629%\">[latex]3\\cdot{x}=+3x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 23.2394%\">[latex]+3[\/latex]<\/td>\r\n<td style=\"width: 38.2629%\">[latex]x\\cdot{3}=+3x[\/latex]<\/td>\r\n<td style=\"width: 38.2629%\">\u00a0[latex]3\\cdot{3}=+9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow we can write the terms of the polynomial from the entries in the table:\r\n\r\n[latex]\\left(x+3\\right)^{2}[\/latex]\r\n\r\n=\u00a0[latex]x^2[\/latex] + [latex]3x[\/latex] + [latex]3x[\/latex] + [latex]9[\/latex]\r\n\r\n= [latex]x^{2}[\/latex] + [latex]6x[\/latex] + [latex]9[\/latex].\r\n\r\nPretty cool, huh?\r\n\r\nPolynomials can take many forms. \u00a0So far we have seen examples of binomials with variable terms on the left and constant terms on the right, such as this binomial [latex]\\left(2r-3\\right)[\/latex]. \u00a0Variables may also be on the right of the constant term, as in this binomial [latex]\\left(5+r\\right)[\/latex]. \u00a0In the next example, we will show that multiplying binomials in this form requires one extra step at the end.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the product.[latex]\\left(3\u2013s\\right)\\left(1-s\\right)[\/latex]\r\n[reveal-answer q=\"531601\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"531601\"]\r\n\r\nNotice how the binomials have the variable on the right instead of the left. \u00a0There is nothing different in the way you find the product. \u00a0At the end we will reorganize terms so they are in descending order as a matter of convention.\r\n\r\n[latex]\\left(3\u2013s\\right)\\left(1\u2013s\\right)[\/latex]\r\n\r\nUse a table this time.\r\n<table style=\"width: 40%\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<th>[latex]3[\/latex]<\/th>\r\n<th>[latex]-s[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>[latex]1[\/latex]<\/th>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]-s[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<th>[latex]-s[\/latex]<\/th>\r\n<td>[latex]-3s[\/latex]<\/td>\r\n<td>[latex]s^2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice how the <em>s<\/em> term is now positive. Collect the terms and simplify.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\left(3\u2013s\\right)\\left(1\u2013s\\right)\\\\\\text{ }\\\\=3-3s-s+s^2\\\\\\text{ }\\\\=3-4s+s^2\\end{array}[\/latex]<\/p>\r\nAs a matter of convention, we will organize the terms so the one with greatest degree comes first. Pay close attention to the signs on the terms when you reorganize them. The [latex]3[\/latex] is positive, so we will use a plus in front of it, and the [latex]4[\/latex] is negative so we use a minus in front of it.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\left(3\u2013s\\right)\\left(1\u2013s\\right)\\\\\\text{ }\\\\=s^{2}-4s+3\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(3\u2013s\\right)\\left(1\u2013s\\right)=s^2-4s+3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Using FOIL to Multiply Binomials<\/h2>\r\n[caption id=\"attachment_4589\" align=\"aligncenter\" width=\"335\"]<img class=\" wp-image-4589\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/03212853\/Screen-Shot-2016-06-03-at-2.08.13-PM-243x300.png\" alt=\"Crane made from aluminum foil\" width=\"335\" height=\"414\" \/> Foil Crane[\/caption]\r\n\r\nRemember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes there are no like terms to combine. Let's look at the last example again and pay particular attention to how we got the four terms.\r\n<p style=\"text-align: center\">[latex]\\left(x+2\\right)\\left(x-y\\right)[\/latex]\r\n[latex]{x}^{2}-\\mathit{\\text{xy}}+2x - 2y[\/latex]<\/p>\r\nWhere did the first term, [latex]{x}^{2}[\/latex], come from?\r\n\r\nIt is the product of [latex]x\\text{ and }x[\/latex], the <strong>first<\/strong> terms in [latex]\\left(x+2\\right)\\text{and}\\left(x-y\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224439\/CNX_BMath_Figure_10_03_016_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a red arrow from the first x to the second. Beside this, \" \/>\r\nThe next term, [latex]-\\mathit{\\text{xy}}[\/latex], is the product of [latex]x\\text{ and }-y[\/latex], the two <strong>outer<\/strong> terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224440\/CNX_BMath_Figure_10_03_017_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a red arrow from the first x to the y. Beside this, \" \/>\r\nThe third term, [latex]+2x[\/latex], is the product of [latex]2\\text{ and }x[\/latex], the two <strong>inner<\/strong> terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224441\/CNX_BMath_Figure_10_03_018_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a black arrow from the first x to the y. There is a red arrow from the 2 to the x. Below that, \" \/>\r\nAnd the last term, [latex]-2y[\/latex], came from multiplying the two <strong>last<\/strong> terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224441\/CNX_BMath_Figure_10_03_019_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a black arrow from the first x to the y. There is a black arrow from the 2 to the x. There is a red arrow from the 2 to the y. Above that, \" \/>\r\n\r\nWe can use a shortcut called the FOIL method when multiplying two binomials.\u00a0Some people use the FOIL method to keep track of which pairs of terms have been multiplied when you are multiplying two binomials. This is not the same thing you use to wrap up leftovers, but an acronym for <strong>First, Outer, Inner, Last.<\/strong>\u00a0It is called FOIL because we multiply the <strong>f<\/strong>irst terms, the <strong>o<\/strong>uter terms, the <strong>i<\/strong>nner terms, and then the <strong>l<\/strong>ast terms of each binomial.\u00a0\u00a0The FOIL method arises out of using the distributive property to multiply two binomials. We are simply multiplying each term of the first binomial by each term of the second binomial and then combining like terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200224\/CNX_CAT_Figure_01_04_003.jpg\" alt=\"Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.\" \/>\r\n\r\nLet's go back to the example [latex]\\left(x+2\\right)\\left(x-y\\right)[\/latex]. \u00a0The following steps show you how to apply this method to multiplying two binomials.\r\n\r\n[latex]\\begin{array}{l}\\text{First}\\text{ term in each binomial}: \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+2\\right)\\left(x-y\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(x\\right)=x^{2}\\\\\\text{Outer terms}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+2\\right)\\left(x-y\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(-y\\right)=-xy\\\\\\text{Inner terms}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+2\\right)\\left(x-y\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\left(x\\right)=2x\\\\\\text{Last terms in each binomial}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+2\\right)\\left(x-y\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\left(-y\\right)=-2y\\end{array}[\/latex]\r\n\r\nWhen you add the four results, you get the same answer,\u00a0[latex]x^{2}-xy+2x-2y[\/latex].\r\n\r\nThe last step in multiplying polynomials is to combine like terms.\u00a0 In this example there were no like terms, but you will see this last step in several of the examples below. Remember that a polynomial is simplified only when there are no like terms remaining.\r\n<div class=\"textbox shaded\"><img class=\" wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"41\" height=\"36\" \/>Caution! Note that the FOIL method only works for multiplying two binomials together. It will not work for multiplying a binomial and a trinomial, or two trinomials.<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse FOIL to find the product. [latex](2x-18)(3x+3)[\/latex]\r\n[reveal-answer q=\"787670\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"787670\"]\r\n\r\nFind the product of the first terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200225\/CNX_CAT_Figure_01_04_004.jpg\" alt=\"\" \/>\r\n\r\nFind the product of the outer terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200227\/CNX_CAT_Figure_01_04_005.jpg\" alt=\"\" \/>\r\n\r\nFind the product of the inner terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200228\/CNX_CAT_Figure_01_04_006.jpg\" alt=\"\" \/>\r\n\r\nFind the product of the last terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200229\/CNX_CAT_Figure_01_04_007.jpg\" alt=\"\" \/>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{cc}6{x}^{2}+6x - 54x - 54\\hfill &amp; \\text{Add the products}.\\hfill \\\\ 6{x}^{2}+\\left(6x - 54x\\right)-54\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 6{x}^{2}-48x - 54\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following steps summarize the process for using FOIL to multiply two binomials. It is very important to note that this process only works for the product of two binomials. If you are multiplying a binomial\u00a0and a trinomial, it is better to use a table to keep track of your terms.\r\n<div class=\"textbox shaded\">\r\n<h3>Use the FOIL method for multiplying two binomials<\/h3>\r\n<ol id=\"eip-id1168469711482\" class=\"stepwise\">\r\n \t<li>Multiply the <strong>First<\/strong> terms.<\/li>\r\n \t<li>Multiply the <strong>Outer<\/strong> terms.<\/li>\r\n \t<li>Multiply the <strong>Inner<\/strong> terms.<\/li>\r\n \t<li>Multiply the <strong>Last<\/strong> terms.<\/li>\r\n \t<li>Combine like terms, when possible.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224442\/CNX_BMath_Figure_10_03_025_img.png\" alt=\"Parentheses a plus b times parentheses c plus d is shown. Above a is first, above b is last, above c is first, above d is last. There is a brace connecting a and d that says outer. There is a brace connecting b and c that says inner.\" width=\"187\" height=\"137\" \/><\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(y - 8\\right)\\left(y+6\\right)[\/latex]\r\n[reveal-answer q=\"120940\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"120940\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466233448\" class=\"unnumbered unstyled\" summary=\"The first line says, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>Step 1<\/strong>: Multiply the <strong>First<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224453\/CNX_BMath_Figure_10_03_055_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 2<\/strong>: Multiply the <strong>Outer<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224455\/CNX_BMath_Figure_10_03_055_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 3<\/strong>: Multiply the <strong>Inner<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224456\/CNX_BMath_Figure_10_03_055_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 4<\/strong>: Multiply the <strong>Last<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224501\/CNX_BMath_Figure_10_03_055_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 5<\/strong>: Combine like terms<\/td>\r\n<td>[latex]y^2-2y-48[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146212[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(2a+3\\right)\\left(3a - 1\\right)[\/latex]\r\n[reveal-answer q=\"807420\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"807420\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468512120\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 2a plus 3 times parentheses 3a minus 1. The next line shows the same thing, but with arrows pointing from the 2a to the 3a, from the 2a to the 1, from the 3 to the 3a, and from the 3 to the 1. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex](2a+3)(3a-1)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224505\/CNX_BMath_Figure_10_03_056_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>First<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224506\/CNX_BMath_Figure_10_03_056_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>Outer<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224507\/CNX_BMath_Figure_10_03_056_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>Inner<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224510\/CNX_BMath_Figure_10_03_056_img-05.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>Last<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224511\/CNX_BMath_Figure_10_03_056_img-06.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]6a^2+7a-3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146213[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(5x-y\\right)\\left(2x - 7\\right)[\/latex]\r\n[reveal-answer q=\"840187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"840187\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468525268\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 5x minus y times parentheses 2x minus 7. The next line shows the same thing, but with arrows pointing from the 5x to the 2x, from the 5x to the 7, from the y to the 2x, and from the y to the 7. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex](5x-y)(2x-7)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224515\/CNX_BMath_Figure_10_03_057_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>First<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224516\/CNX_BMath_Figure_10_03_057_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>Outer<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224517\/CNX_BMath_Figure_10_03_057_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>Inner<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224518\/CNX_BMath_Figure_10_03_057_img-05.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>Last<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224519\/CNX_BMath_Figure_10_03_057_img-06.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms. There are none.<\/td>\r\n<td>[latex]10x^2-35x-2xy+7y[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146215[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn the following video, we show an example of how to use the FOIL method to multiply two binomials.\r\n\r\nhttps:\/\/youtu.be\/_MrdEFnXNGA\r\nFor another example of using the FOIL method to multiply two binomials watch the next video.\r\n\r\nhttps:\/\/youtu.be\/0HzsAjucUaw\r\n<h2>Multiplying Two Binomials Using the Vertical Method<\/h2>\r\nThe FOIL method is usually the quickest method for multiplying two binomials, but it works <em>only<\/em> for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224522\/CNX_BMath_Figure_10_03_058_img.png\" alt=\"A vertical multiplication problem is shown. 23 times 46 is written with a line underneath. Beneath the line is 138. Beside 138 is written \" \/>\r\nYou start by multiplying [latex]23[\/latex] by [latex]6[\/latex] to get [latex]138[\/latex].\r\n\r\nThen you multiply [latex]23[\/latex] by [latex]4[\/latex], lining up the partial product in the correct columns.\r\n\r\nLast, you add the partial products.\r\n\r\nNow we'll apply this same method to multiply two binomials.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply using the vertical method: [latex]\\left(5x - 1\\right)\\left(2x - 7\\right)[\/latex]\r\n\r\n[reveal-answer q=\"985210\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"985210\"]\r\n\r\nSolution\r\nIt does not matter which binomial goes on the top. Line up the columns when you multiply as we did when we multiplied [latex]23\\left(46\\right)[\/latex].\r\n<table id=\"eip-id1168469481295\" class=\"unnumbered unstyled\" summary=\"A vertical multiplication problem is shown. 2x minus 7 times 5x minus 1 is written with a line underneath. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224523\/CNX_BMath_Figure_10_03_059_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]2x - 7[\/latex] by [latex]-1[\/latex] .<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224524\/CNX_BMath_Figure_10_03_059_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]2x - 7[\/latex] by [latex]5x[\/latex] .<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224526\/CNX_BMath_Figure_10_03_059_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add like terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224527\/CNX_BMath_Figure_10_03_059_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice the partial products are the same as the terms in the FOIL method.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224528\/CNX_BMath_Figure_10_03_060_img.png\" alt=\"On the left, 5x minus 1 times 2x minus 7 is shown. Below that is 10 x squared minus 35x minus 2x plus 7. The first two terms are in blue, the second two in red. Beneath that is 10 x squared minus 37x plus 7. On the right, a vertical multiplication problem is shown. 2xx minus 7 times 5x minus 1 is written with a line underneath. Beneath the line is a red negative 2x plus 7. Beneath that is 10 x squared minus 35 x in blue. Beneath that, there is another line. Beneath that line is 10 x squared minus 37x plus 7.\" \/>[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146216[\/ohm_question]\r\n\r\n<\/div>\r\n\r\n[caption id=\"attachment_4595\" align=\"alignleft\" width=\"138\"]<img class=\" wp-image-4595\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/03231013\/Screen-Shot-2016-06-03-at-4.09.32-PM-299x300.png\" alt=\"two tomatoes sitting next to each other with two different phonetic pronunciations for the word tomato underneath\" width=\"138\" height=\"139\" \/> Order Doesn't Matter When You Multiply[\/caption]\r\n\r\nOne of the neat things about multiplication is that\u00a0terms can be multiplied in either order. The expression [latex]\\left(x+2\\right)\\left(x+4\\right)[\/latex] has the same product as [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex], [latex]x^{2}+6x+8[\/latex]. (Work it out and see.) The order in which you multiply binomials does not matter. What matters is that you multiply each term in one binomial by each term in the other binomial.\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nLet's look at another example using the FOIL method in which the variables have coefficients.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]\\left(4x\u201310\\right)\\left(2x+3\\right)[\/latex] using the FOIL acronym.\r\n\r\n[reveal-answer q=\"930433\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"930433\"]\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,4x\\left(2x\\right)=8x^{2}\\\\\\text{Outer}:\\,\\,\\,4x\\left(3\\right)=12x\\\\\\text{Inner}:\\,\\,\\,\u221210\\left(2x\\right)=-20x\\\\\\text{Last}:\\,\\,\\,\\,\\,-10\\left(3\\right)=-30\\end{array}[\/latex]<\/p>\r\nBe careful about including the negative sign on the [latex]\u201110[\/latex], since 10 is subtracted.\r\n\r\nCombine like terms.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}8x^{2}+12x\u201320x\u201330\\\\\\text{ }\\\\=8x^{2}-8x\u201330\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(4x\u201310\\right)\\left(2x+3\\right)=8x^{2}\u20138x\u201330[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nSo far, we have shown several methods for multiplying two binomials together. \u00a0Why are we focusing\u00a0so much on binomials? \u00a0They are one of the most well studied and widely used polynomials, so there is a lot of information out there about them. In the previous example, we saw the result of squaring a binomial that was a sum of two terms. In the next example we will find the product of squaring a binomial that is the difference of two terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSquare the binomial difference\u00a0[latex]\\left(x\u20137\\right)[\/latex]\r\n[reveal-answer q=\"293164\"]Show solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"293164\"]\r\n\r\nWrite the product of the binomial.\r\n<p style=\"text-align: center\">[latex]{\\left(x-7\\right)}^2=\\left(x\u20137\\right)\\left(x\u20137\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left\">Let's use the table method, just because. Note how we carry the negative sign with the\u00a0[latex]7[\/latex].<\/p>\r\n\r\n<table style=\"width: 20%\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\u00a0[latex]x[\/latex]<\/td>\r\n<td>\u00a0[latex]-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]x[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]x^2[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]-7x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0\u00a0[latex]-7[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]-7x[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]49[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCollect the terms, and simplify. Note how we keep the sign on each term.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}x^2-7x-7x+49\\\\\\text{ }\\\\=x^2-14x+49\\end{array}[\/latex]<\/p>\r\nAnswer\r\n[latex]x^2-14x+49[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"51\" height=\"45\" \/>Caution! It is VERY important to remember the caution from the exponents section about squaring a binomial:\r\n<p style=\"text-align: center\">You can't move the exponent into a grouped sum because of the order of operations!!!!!<\/p>\r\n<p style=\"text-align: center\"><strong>INCORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}\\neq2^{2}+x^{2}[\/latex]<\/p>\r\n<p style=\"text-align: center\"><strong>\u00a0CORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}=\\left(2+x\\right)\\left(2+x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\nIn the video that follows, you will see another examples of using a table to multiply two binomials.\r\n\r\nhttps:\/\/youtu.be\/tWsLJ_pn5mQ\r\n<h2>Further Examples<\/h2>\r\nThe next couple of examples show you some different forms binomials can take. \u00a0In the first, we will square a binomial that has a coefficient in front of the variable, like the product in the first example on this page. In the second we will find\u00a0the product of two binomials that have the variable on the right instead of the left.\u00a0We will use both\u00a0the FOIL method and the table method to simplify.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the product.\u00a0[latex]\\left(2x+6\\right)^{2}[\/latex]\r\n[reveal-answer q=\"255359\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"255359\"]We will use the FOIL method.\r\n[latex]\\left(2x+6\\right)^{2}=\\left(2x+6\\right)\\left(2x+6\\right)[\/latex]\r\n[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,2x\\left(2x\\right)=4x^{2}\\\\\\text{Outer}:\\,\\,\\,2x\\left(6\\right)=12x\\\\\\text{Inner}:\\,\\,\\,6\\left(2x\\right)=12x\\\\\\text{Last}:\\,\\,\\,\\,\\,6\\left(6\\right)=36\\end{array}[\/latex]\r\n\r\nNow you can collect the terms and simplify:\r\n[latex]\\begin{array}{c}4x^2+12x+12x+36\\\\\\text{ }\\\\=4x^2+24x+36\\end{array}[\/latex]\r\n\r\nAnswer\r\n\r\n[latex](2x+6)^{2}=4x^{2}+24x+36[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the last example, we want to show you another common form a binomial can take, each of the terms in the two binomials is the same, but the signs are different. You will see that in this case, the middle term will disappear.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply the binomials. [latex]\\left(x+8\\right)\\left(x\u20138\\right)[\/latex]\r\n[reveal-answer q=\"812247\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"812247\"]\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(x\\right)=x^{2}\\\\\\text{Outer}:\\,\\,\\,\\,\\,\\,x\\left(-8\\right)=-8x\\\\\\text{Inner}:\\,\\,\\,\\,\\,\\,\\,8\\left(x\\right)=+8x\\\\\\text{Last}:\\,\\,\\,\\,\\,\\,\\,\\,\\,8\\left(-8\\right)=-64\\end{array}[\/latex]<\/p>\r\nAdd the terms. Note how the two x terms are opposites, so they sum to zero.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}x^{2}\\underbrace{-8x+8x}-64\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\text{ }\\\\=x^2-64\\\\\\text{ }\\\\\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(x+8\\right)\\left(x-8\\right)=x^{2}-64[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\n<p style=\"text-align: left\">There are predictable outcomes when you square a binomial sum or difference. In general terms, for a binomial difference,<\/p>\r\n<p style=\"text-align: center\">[latex]\\left(a-b\\right)^{2}=\\left(a-b\\right)\\left(a-b\\right)[\/latex],<\/p>\r\n<p style=\"text-align: left\">the resulting product, after being simplified, will look like this:<\/p>\r\n<p style=\"text-align: center\">[latex]a^2-2ab+b^2[\/latex].<\/p>\r\n<p style=\"text-align: left\">The product of a binomial sum will have the following predictable outcome:<\/p>\r\n<p style=\"text-align: center\">[latex]\\left(a+b\\right)^{2}=\\left(a+b\\right)\\left(a+b\\right)=a^2+2ab+b^2[\/latex].<\/p>\r\nThe\u00a0product of a binomial sum and binomial difference of the same two monomial will have the following predictable outcome:\r\n<p style=\"text-align: center\">[latex]\\left(a+b\\right)\\left(a-b\\right)=a^2-b^2[\/latex].<\/p>\r\n<p style=\"text-align: left\">Note that a and b in these generalizations could be integers, fractions, or variables with any kind of constant. \u00a0You will learn more about predictable patterns from products of binomials in later math classes.<\/p>\r\n\r\n<\/div>\r\nIn this section we showed how to multiply two binomials using the distributive property, an area model, by using a table, using the FOIL method, and the vertical method.\u00a0 Practice each method, and try to decide which one you prefer.\r\n\r\nSome of the forms a product of two binomials can take are listed here:\r\n<ul>\r\n \t<li>[latex]\\left(x+5\\right)\\left(2x-3\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(x+7\\right)^{2}[\/latex]<\/li>\r\n \t<li>[latex]\\left(x-1\\right)^{2}[\/latex]<\/li>\r\n \t<li>[latex]\\left(2-y\\right)\\left(5+y\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(x+9\\right)\\left(x-9\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(2x-4\\right)\\left(x+3\\right)[\/latex]<\/li>\r\n<\/ul>\r\nAnd this is just a small list, the possible combinations are endless. For each of the products in the list, using one of the two methods presented here will work to simplify.\r\n<h2>Summary<\/h2>\r\nMultiplication of binomials and polynomials requires an understanding of the distributive property, rules for exponents, and a keen eye for collecting like terms. Whether the polynomials are monomials, binomials, or trinomials, carefully multiply each term in one polynomial by each term in the other polynomial. Be careful to watch the addition and subtraction signs and negative coefficients. A product is written in simplified form if all of its like terms have been combined.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Multiply binomials<\/li>\n<\/ul>\n<\/div>\n<p>Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial.<\/p>\n<h3>Using the Distributive Property<\/h3>\n<p>We will start by using the Distributive Property. Look again at the following example.<\/p>\n<table id=\"eip-id1168466233256\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x plus 3, times p, with red arrows from the p to the x and to the 3. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224428\/CNX_BMath_Figure_10_03_049_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>We distributed the [latex]p[\/latex] to get<\/td>\n<td>[latex]x\\color{red}{p}+3\\color{red}{p}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>What if we have [latex]\\left(x+7\\right)[\/latex] instead of [latex]p[\/latex] ?<\/p>\n<p>Think of the [latex]x+7[\/latex] as the [latex]\\color{red}{p}[\/latex] above.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224430\/CNX_BMath_Figure_10_03_049_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute [latex]\\left(x+7\\right)[\/latex] .<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224432\/CNX_BMath_Figure_10_03_049_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute again.<\/td>\n<td>[latex]{x}^{2}+7x+3x+21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]{x}^{2}+10x+21[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that before combining like terms, we had four terms. We multiplied the two terms of the first binomial by the two terms of the second binomial\u2014four multiplications.<\/p>\n<p>Be careful to distinguish between a sum and a product.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{cccc}\\hfill \\mathbf{\\text{Sum}}\\hfill & & & \\hfill \\mathbf{\\text{Product}}\\hfill \\\\ \\hfill x+x\\hfill & & & \\hfill x\\cdot x\\hfill \\\\ \\hfill 2x\\hfill & & & \\hfill {x}^{2}\\hfill \\\\ \\hfill \\text{combine like terms}\\hfill & & & \\hfill \\text{add exponents of like bases}\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(x+6\\right)\\left(x+8\\right)[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168468281692\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x plus 6 times parentheses x plus 8. The next line shows parentheses x plus 6 times red parentheses x plus 8, with red arrows from x plus 8 to x and to 6. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(x+6\\right)\\left(x+8\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224433\/CNX_BMath_Figure_10_03_050_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute [latex]\\left(x+8\\right)[\/latex] .<\/td>\n<td>[latex]x\\color{red}{(x+8)}+6\\color{red}{(x+8)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute again.<\/td>\n<td>[latex]{x}^{2}+8x+6x+48[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{x}^{2}+14x+48[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146207\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146207&theme=oea&iframe_resize_id=ohm146207&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Now we&#8217;ll see how to multiply binomials where the variable has a coefficient.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(2x+9\\right)\\left(3x+4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q901421\">Show Solution<\/span><\/p>\n<div id=\"q901421\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467332174\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 2x plus 9 times parentheses 3x plus 4. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(2x+9\\right)\\left(3x+4\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute. [latex]\\left(3x+4\\right)[\/latex]<\/td>\n<td>[latex]2x\\color{red}{(3x+4)}+9\\color{red}{(3x+4)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute again.<\/td>\n<td>[latex]6{x}^{2}+8x+27x+36[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]6{x}^{2}+35x+36[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146208\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146208&theme=oea&iframe_resize_id=ohm146208&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the previous examples, the binomials were sums. When there are differences, we pay special attention to make sure the signs of the product are correct.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(4y+3\\right)\\left(6y - 5\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q834420\">Show Solution<\/span><\/p>\n<div id=\"q834420\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469625351\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 4y plus 3 times parentheses 6y minus 5. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(4y+3\\right)\\left(6y - 5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]4y\\color{red}{(6y-5)}+3\\color{red}{(6y-5)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute again.<\/td>\n<td>[latex]24{y}^{2}-20y+18y - 15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]24{y}^{2}-2y - 15[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146209\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146209&theme=oea&iframe_resize_id=ohm146209&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Up to this point, the product of two binomials has been a trinomial. This is not always the case.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(x+2\\right)\\left(x-y\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q982155\">Show Solution<\/span><\/p>\n<div id=\"q982155\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468254725\" class=\"unnumbered unstyled\" summary=\"The top line says parentheses x plus 2 times parentheses x minus y. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex](x+2)(x-y)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]x\\color{red}{(x-y)}+2\\color{red}{(x-y)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute again.<\/td>\n<td>[latex]x^2-xy+2x-2y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>There are no like terms to combine.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146210\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146210&theme=oea&iframe_resize_id=ohm146210&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h3>Area Model for Multiplying Binomials<\/h3>\n<p>Now let&#8217;s explore multiplying two binomials. For those of you that use pictures to learn, you can draw an area model to help make sense of the process. You&#8217;ll use each binomial as one of the dimensions of a rectangle, and their product as the area.<\/p>\n<p>The model below shows [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]:<\/p>\n<div id=\"attachment_4607\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4607\" class=\"size-medium wp-image-4607\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/04191552\/Screen-Shot-2016-06-04-at-12.15.06-PM-300x290.png\" alt=\"Visual representation of multiplying two binomials.\" width=\"300\" height=\"290\" \/><\/p>\n<p id=\"caption-attachment-4607\" class=\"wp-caption-text\">Visual representation of multiplying two binomials.<\/p>\n<\/div>\n<p>Each binomial is expanded into variable terms and constants, [latex]x+4[\/latex], along the top of the model and [latex]x+2[\/latex] along the left side. The product of each pair of terms is a colored rectangle. The total area is the sum of all of these small rectangles, [latex]x^{2}+2x+4x+8[\/latex], If you combine all the like terms, you can write the product, or area, as [latex]x^{2}+6x+8[\/latex].<\/p>\n<p>You can use the distributive property to determine the product of two binomials.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q186797\">Show Solution<\/span><\/p>\n<div id=\"q186797\" class=\"hidden-answer\" style=\"display: none\">Distribute the [latex]x[\/latex] over [latex]x+2[\/latex], then distribute 4 over [latex]x+2[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]x\\left(x\\right)+x\\left(2\\right)+4\\left(x\\right)+4\\left(2\\right)[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center\">[latex]x^{2}+2x+4x+8[\/latex]<\/p>\n<p>Combine like terms [latex]\\left(2x+4x\\right)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]x^{2}+6x+8[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(x+4\\right)\\left(2x+2\\right)=x^{2}+6x+8[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Look back at the model above to see where each piece of [latex]x^{2}+2x+4x+8[\/latex] comes from. Can you see where you multiply [latex]x[\/latex] by [latex]x + 2[\/latex], and where you get [latex]x^{2}[\/latex]\u00a0from [latex]x\\left(x\\right)[\/latex]?<\/p>\n<p>Another way to look at multiplying binomials is to see that each term in one binomial is multiplied by each term in the other binomial. Look at the example above: the [latex]x[\/latex] in [latex]x+4[\/latex] gets multiplied by both the [latex]x[\/latex] and the [latex]2[\/latex] from [latex]x+2[\/latex], and the [latex]4[\/latex] gets multiplied by both the [latex]x[\/latex] and the [latex]2[\/latex].<\/p>\n<p>The following video provides an example of multiplying two binomials using an area model as well as repeated distribution.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Multiply Binomials Using An Area Model and Using Repeated Distribution\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/u4Hgl0BrUlo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>The Table Method<\/h2>\n<p>You may see a binomial multiplied by itself written as\u00a0[latex]{\\left(x+3\\right)}^{2}[\/latex] instead of\u00a0[latex]\\left(x+3\\right)\\left(x+3\\right)[\/latex]. \u00a0To find this product, let&#8217;s use another method. We will place the terms of each binomial along the top row and first column of a table, like this:<\/p>\n<table style=\"width: 50%\">\n<thead>\n<tr>\n<td style=\"width: 23.2394%\"><\/td>\n<td style=\"width: 38.2629%\">[latex]x[\/latex]<\/td>\n<td style=\"width: 38.2629%\">[latex]+3[\/latex]<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 23.2394%\">[latex]x[\/latex]<\/td>\n<td style=\"width: 38.2629%\"><\/td>\n<td style=\"width: 38.2629%\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 23.2394%\">[latex]+3[\/latex]<\/td>\n<td style=\"width: 38.2629%\"><\/td>\n<td style=\"width: 38.2629%\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now multiply the term in each column by the term in each row to get the terms of the resulting polynomial. Note how we keep the signs on the terms, even when they are positive, this will help us write the new polynomial.<\/p>\n<table style=\"width: 50%\">\n<thead>\n<tr>\n<td style=\"width: 23.2394%\"><\/td>\n<td style=\"width: 38.2629%\">[latex]x[\/latex]<\/td>\n<td style=\"width: 38.2629%\">[latex]+3[\/latex]<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 23.2394%\">[latex]x[\/latex]<\/td>\n<td style=\"width: 38.2629%\">[latex]x\\cdot{x}=x^2[\/latex]<\/td>\n<td style=\"width: 38.2629%\">[latex]3\\cdot{x}=+3x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 23.2394%\">[latex]+3[\/latex]<\/td>\n<td style=\"width: 38.2629%\">[latex]x\\cdot{3}=+3x[\/latex]<\/td>\n<td style=\"width: 38.2629%\">\u00a0[latex]3\\cdot{3}=+9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now we can write the terms of the polynomial from the entries in the table:<\/p>\n<p>[latex]\\left(x+3\\right)^{2}[\/latex]<\/p>\n<p>=\u00a0[latex]x^2[\/latex] + [latex]3x[\/latex] + [latex]3x[\/latex] + [latex]9[\/latex]<\/p>\n<p>= [latex]x^{2}[\/latex] + [latex]6x[\/latex] + [latex]9[\/latex].<\/p>\n<p>Pretty cool, huh?<\/p>\n<p>Polynomials can take many forms. \u00a0So far we have seen examples of binomials with variable terms on the left and constant terms on the right, such as this binomial [latex]\\left(2r-3\\right)[\/latex]. \u00a0Variables may also be on the right of the constant term, as in this binomial [latex]\\left(5+r\\right)[\/latex]. \u00a0In the next example, we will show that multiplying binomials in this form requires one extra step at the end.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the product.[latex]\\left(3\u2013s\\right)\\left(1-s\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q531601\">Show Solution<\/span><\/p>\n<div id=\"q531601\" class=\"hidden-answer\" style=\"display: none\">\n<p>Notice how the binomials have the variable on the right instead of the left. \u00a0There is nothing different in the way you find the product. \u00a0At the end we will reorganize terms so they are in descending order as a matter of convention.<\/p>\n<p>[latex]\\left(3\u2013s\\right)\\left(1\u2013s\\right)[\/latex]<\/p>\n<p>Use a table this time.<\/p>\n<table style=\"width: 40%\">\n<tbody>\n<tr>\n<td><\/td>\n<th>[latex]3[\/latex]<\/th>\n<th>[latex]-s[\/latex]<\/th>\n<\/tr>\n<tr>\n<th>[latex]1[\/latex]<\/th>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]-s[\/latex]<\/td>\n<\/tr>\n<tr>\n<th>[latex]-s[\/latex]<\/th>\n<td>[latex]-3s[\/latex]<\/td>\n<td>[latex]s^2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice how the <em>s<\/em> term is now positive. Collect the terms and simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\left(3\u2013s\\right)\\left(1\u2013s\\right)\\\\\\text{ }\\\\=3-3s-s+s^2\\\\\\text{ }\\\\=3-4s+s^2\\end{array}[\/latex]<\/p>\n<p>As a matter of convention, we will organize the terms so the one with greatest degree comes first. Pay close attention to the signs on the terms when you reorganize them. The [latex]3[\/latex] is positive, so we will use a plus in front of it, and the [latex]4[\/latex] is negative so we use a minus in front of it.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\left(3\u2013s\\right)\\left(1\u2013s\\right)\\\\\\text{ }\\\\=s^{2}-4s+3\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(3\u2013s\\right)\\left(1\u2013s\\right)=s^2-4s+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Using FOIL to Multiply Binomials<\/h2>\n<div id=\"attachment_4589\" style=\"width: 345px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4589\" class=\"wp-image-4589\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/03212853\/Screen-Shot-2016-06-03-at-2.08.13-PM-243x300.png\" alt=\"Crane made from aluminum foil\" width=\"335\" height=\"414\" \/><\/p>\n<p id=\"caption-attachment-4589\" class=\"wp-caption-text\">Foil Crane<\/p>\n<\/div>\n<p>Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes there are no like terms to combine. Let&#8217;s look at the last example again and pay particular attention to how we got the four terms.<\/p>\n<p style=\"text-align: center\">[latex]\\left(x+2\\right)\\left(x-y\\right)[\/latex]<br \/>\n[latex]{x}^{2}-\\mathit{\\text{xy}}+2x - 2y[\/latex]<\/p>\n<p>Where did the first term, [latex]{x}^{2}[\/latex], come from?<\/p>\n<p>It is the product of [latex]x\\text{ and }x[\/latex], the <strong>first<\/strong> terms in [latex]\\left(x+2\\right)\\text{and}\\left(x-y\\right)[\/latex].<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224439\/CNX_BMath_Figure_10_03_016_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a red arrow from the first x to the second. Beside this,\" \/><br \/>\nThe next term, [latex]-\\mathit{\\text{xy}}[\/latex], is the product of [latex]x\\text{ and }-y[\/latex], the two <strong>outer<\/strong> terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224440\/CNX_BMath_Figure_10_03_017_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a red arrow from the first x to the y. Beside this,\" \/><br \/>\nThe third term, [latex]+2x[\/latex], is the product of [latex]2\\text{ and }x[\/latex], the two <strong>inner<\/strong> terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224441\/CNX_BMath_Figure_10_03_018_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a black arrow from the first x to the y. There is a red arrow from the 2 to the x. Below that,\" \/><br \/>\nAnd the last term, [latex]-2y[\/latex], came from multiplying the two <strong>last<\/strong> terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224441\/CNX_BMath_Figure_10_03_019_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a black arrow from the first x to the y. There is a black arrow from the 2 to the x. There is a red arrow from the 2 to the y. Above that,\" \/><\/p>\n<p>We can use a shortcut called the FOIL method when multiplying two binomials.\u00a0Some people use the FOIL method to keep track of which pairs of terms have been multiplied when you are multiplying two binomials. This is not the same thing you use to wrap up leftovers, but an acronym for <strong>First, Outer, Inner, Last.<\/strong>\u00a0It is called FOIL because we multiply the <strong>f<\/strong>irst terms, the <strong>o<\/strong>uter terms, the <strong>i<\/strong>nner terms, and then the <strong>l<\/strong>ast terms of each binomial.\u00a0\u00a0The FOIL method arises out of using the distributive property to multiply two binomials. We are simply multiplying each term of the first binomial by each term of the second binomial and then combining like terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200224\/CNX_CAT_Figure_01_04_003.jpg\" alt=\"Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.\" \/><\/p>\n<p>Let&#8217;s go back to the example [latex]\\left(x+2\\right)\\left(x-y\\right)[\/latex]. \u00a0The following steps show you how to apply this method to multiplying two binomials.<\/p>\n<p>[latex]\\begin{array}{l}\\text{First}\\text{ term in each binomial}: \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+2\\right)\\left(x-y\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(x\\right)=x^{2}\\\\\\text{Outer terms}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+2\\right)\\left(x-y\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(-y\\right)=-xy\\\\\\text{Inner terms}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+2\\right)\\left(x-y\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\left(x\\right)=2x\\\\\\text{Last terms in each binomial}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+2\\right)\\left(x-y\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\left(-y\\right)=-2y\\end{array}[\/latex]<\/p>\n<p>When you add the four results, you get the same answer,\u00a0[latex]x^{2}-xy+2x-2y[\/latex].<\/p>\n<p>The last step in multiplying polynomials is to combine like terms.\u00a0 In this example there were no like terms, but you will see this last step in several of the examples below. Remember that a polynomial is simplified only when there are no like terms remaining.<\/p>\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"41\" height=\"36\" \/>Caution! Note that the FOIL method only works for multiplying two binomials together. It will not work for multiplying a binomial and a trinomial, or two trinomials.<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use FOIL to find the product. [latex](2x-18)(3x+3)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q787670\">Show Solution<\/span><\/p>\n<div id=\"q787670\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the product of the first terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200225\/CNX_CAT_Figure_01_04_004.jpg\" alt=\"\" \/><\/p>\n<p>Find the product of the outer terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200227\/CNX_CAT_Figure_01_04_005.jpg\" alt=\"\" \/><\/p>\n<p>Find the product of the inner terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200228\/CNX_CAT_Figure_01_04_006.jpg\" alt=\"\" \/><\/p>\n<p>Find the product of the last terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200229\/CNX_CAT_Figure_01_04_007.jpg\" alt=\"\" \/><\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{cc}6{x}^{2}+6x - 54x - 54\\hfill & \\text{Add the products}.\\hfill \\\\ 6{x}^{2}+\\left(6x - 54x\\right)-54\\hfill & \\text{Combine like terms}.\\hfill \\\\ 6{x}^{2}-48x - 54\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following steps summarize the process for using FOIL to multiply two binomials. It is very important to note that this process only works for the product of two binomials. If you are multiplying a binomial\u00a0and a trinomial, it is better to use a table to keep track of your terms.<\/p>\n<div class=\"textbox shaded\">\n<h3>Use the FOIL method for multiplying two binomials<\/h3>\n<ol id=\"eip-id1168469711482\" class=\"stepwise\">\n<li>Multiply the <strong>First<\/strong> terms.<\/li>\n<li>Multiply the <strong>Outer<\/strong> terms.<\/li>\n<li>Multiply the <strong>Inner<\/strong> terms.<\/li>\n<li>Multiply the <strong>Last<\/strong> terms.<\/li>\n<li>Combine like terms, when possible.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224442\/CNX_BMath_Figure_10_03_025_img.png\" alt=\"Parentheses a plus b times parentheses c plus d is shown. Above a is first, above b is last, above c is first, above d is last. There is a brace connecting a and d that says outer. There is a brace connecting b and c that says inner.\" width=\"187\" height=\"137\" \/><\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(y - 8\\right)\\left(y+6\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q120940\">Show Solution<\/span><\/p>\n<div id=\"q120940\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466233448\" class=\"unnumbered unstyled\" summary=\"The first line says,\">\n<tbody>\n<tr>\n<td><strong>Step 1<\/strong>: Multiply the <strong>First<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224453\/CNX_BMath_Figure_10_03_055_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 2<\/strong>: Multiply the <strong>Outer<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224455\/CNX_BMath_Figure_10_03_055_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 3<\/strong>: Multiply the <strong>Inner<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224456\/CNX_BMath_Figure_10_03_055_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 4<\/strong>: Multiply the <strong>Last<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224501\/CNX_BMath_Figure_10_03_055_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 5<\/strong>: Combine like terms<\/td>\n<td>[latex]y^2-2y-48[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146212\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146212&theme=oea&iframe_resize_id=ohm146212&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(2a+3\\right)\\left(3a - 1\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q807420\">Show Solution<\/span><\/p>\n<div id=\"q807420\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468512120\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 2a plus 3 times parentheses 3a minus 1. The next line shows the same thing, but with arrows pointing from the 2a to the 3a, from the 2a to the 1, from the 3 to the 3a, and from the 3 to the 1. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex](2a+3)(3a-1)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224505\/CNX_BMath_Figure_10_03_056_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>First<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224506\/CNX_BMath_Figure_10_03_056_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>Outer<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224507\/CNX_BMath_Figure_10_03_056_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>Inner<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224510\/CNX_BMath_Figure_10_03_056_img-05.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>Last<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224511\/CNX_BMath_Figure_10_03_056_img-06.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]6a^2+7a-3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146213\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146213&theme=oea&iframe_resize_id=ohm146213&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(5x-y\\right)\\left(2x - 7\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q840187\">Show Solution<\/span><\/p>\n<div id=\"q840187\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468525268\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 5x minus y times parentheses 2x minus 7. The next line shows the same thing, but with arrows pointing from the 5x to the 2x, from the 5x to the 7, from the y to the 2x, and from the y to the 7. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex](5x-y)(2x-7)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224515\/CNX_BMath_Figure_10_03_057_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>First<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224516\/CNX_BMath_Figure_10_03_057_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>Outer<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224517\/CNX_BMath_Figure_10_03_057_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>Inner<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224518\/CNX_BMath_Figure_10_03_057_img-05.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>Last<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224519\/CNX_BMath_Figure_10_03_057_img-06.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Combine like terms. There are none.<\/td>\n<td>[latex]10x^2-35x-2xy+7y[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146215\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146215&theme=oea&iframe_resize_id=ohm146215&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the following video, we show an example of how to use the FOIL method to multiply two binomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Multiply Binomials Using the FOIL Acronym\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/_MrdEFnXNGA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\nFor another example of using the FOIL method to multiply two binomials watch the next video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Find The Product of Two Binomials  (09x-52)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/0HzsAjucUaw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Multiplying Two Binomials Using the Vertical Method<\/h2>\n<p>The FOIL method is usually the quickest method for multiplying two binomials, but it works <em>only<\/em> for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224522\/CNX_BMath_Figure_10_03_058_img.png\" alt=\"A vertical multiplication problem is shown. 23 times 46 is written with a line underneath. Beneath the line is 138. Beside 138 is written\" \/><br \/>\nYou start by multiplying [latex]23[\/latex] by [latex]6[\/latex] to get [latex]138[\/latex].<\/p>\n<p>Then you multiply [latex]23[\/latex] by [latex]4[\/latex], lining up the partial product in the correct columns.<\/p>\n<p>Last, you add the partial products.<\/p>\n<p>Now we&#8217;ll apply this same method to multiply two binomials.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply using the vertical method: [latex]\\left(5x - 1\\right)\\left(2x - 7\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q985210\">Show Solution<\/span><\/p>\n<div id=\"q985210\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nIt does not matter which binomial goes on the top. Line up the columns when you multiply as we did when we multiplied [latex]23\\left(46\\right)[\/latex].<\/p>\n<table id=\"eip-id1168469481295\" class=\"unnumbered unstyled\" summary=\"A vertical multiplication problem is shown. 2x minus 7 times 5x minus 1 is written with a line underneath. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224523\/CNX_BMath_Figure_10_03_059_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]2x - 7[\/latex] by [latex]-1[\/latex] .<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224524\/CNX_BMath_Figure_10_03_059_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]2x - 7[\/latex] by [latex]5x[\/latex] .<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224526\/CNX_BMath_Figure_10_03_059_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add like terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224527\/CNX_BMath_Figure_10_03_059_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice the partial products are the same as the terms in the FOIL method.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224528\/CNX_BMath_Figure_10_03_060_img.png\" alt=\"On the left, 5x minus 1 times 2x minus 7 is shown. Below that is 10 x squared minus 35x minus 2x plus 7. The first two terms are in blue, the second two in red. Beneath that is 10 x squared minus 37x plus 7. On the right, a vertical multiplication problem is shown. 2xx minus 7 times 5x minus 1 is written with a line underneath. Beneath the line is a red negative 2x plus 7. Beneath that is 10 x squared minus 35 x in blue. Beneath that, there is another line. Beneath that line is 10 x squared minus 37x plus 7.\" \/><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146216\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146216&theme=oea&iframe_resize_id=ohm146216&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"attachment_4595\" style=\"width: 148px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4595\" class=\"wp-image-4595\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/03231013\/Screen-Shot-2016-06-03-at-4.09.32-PM-299x300.png\" alt=\"two tomatoes sitting next to each other with two different phonetic pronunciations for the word tomato underneath\" width=\"138\" height=\"139\" \/><\/p>\n<p id=\"caption-attachment-4595\" class=\"wp-caption-text\">Order Doesn&#8217;t Matter When You Multiply<\/p>\n<\/div>\n<p>One of the neat things about multiplication is that\u00a0terms can be multiplied in either order. The expression [latex]\\left(x+2\\right)\\left(x+4\\right)[\/latex] has the same product as [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex], [latex]x^{2}+6x+8[\/latex]. (Work it out and see.) The order in which you multiply binomials does not matter. What matters is that you multiply each term in one binomial by each term in the other binomial.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>Let&#8217;s look at another example using the FOIL method in which the variables have coefficients.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]\\left(4x\u201310\\right)\\left(2x+3\\right)[\/latex] using the FOIL acronym.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q930433\">Show Solution<\/span><\/p>\n<div id=\"q930433\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,4x\\left(2x\\right)=8x^{2}\\\\\\text{Outer}:\\,\\,\\,4x\\left(3\\right)=12x\\\\\\text{Inner}:\\,\\,\\,\u221210\\left(2x\\right)=-20x\\\\\\text{Last}:\\,\\,\\,\\,\\,-10\\left(3\\right)=-30\\end{array}[\/latex]<\/p>\n<p>Be careful about including the negative sign on the [latex]\u201110[\/latex], since 10 is subtracted.<\/p>\n<p>Combine like terms.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}8x^{2}+12x\u201320x\u201330\\\\\\text{ }\\\\=8x^{2}-8x\u201330\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(4x\u201310\\right)\\left(2x+3\\right)=8x^{2}\u20138x\u201330[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>So far, we have shown several methods for multiplying two binomials together. \u00a0Why are we focusing\u00a0so much on binomials? \u00a0They are one of the most well studied and widely used polynomials, so there is a lot of information out there about them. In the previous example, we saw the result of squaring a binomial that was a sum of two terms. In the next example we will find the product of squaring a binomial that is the difference of two terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Square the binomial difference\u00a0[latex]\\left(x\u20137\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q293164\">Show solution<\/span><\/p>\n<div id=\"q293164\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write the product of the binomial.<\/p>\n<p style=\"text-align: center\">[latex]{\\left(x-7\\right)}^2=\\left(x\u20137\\right)\\left(x\u20137\\right)[\/latex]<\/p>\n<p style=\"text-align: left\">Let&#8217;s use the table method, just because. Note how we carry the negative sign with the\u00a0[latex]7[\/latex].<\/p>\n<table style=\"width: 20%\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\u00a0[latex]x[\/latex]<\/td>\n<td>\u00a0[latex]-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]x[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]x^2[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]-7x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0\u00a0[latex]-7[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]-7x[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]49[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Collect the terms, and simplify. Note how we keep the sign on each term.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}x^2-7x-7x+49\\\\\\text{ }\\\\=x^2-14x+49\\end{array}[\/latex]<\/p>\n<p>Answer<br \/>\n[latex]x^2-14x+49[\/latex]\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"51\" height=\"45\" \/>Caution! It is VERY important to remember the caution from the exponents section about squaring a binomial:<\/p>\n<p style=\"text-align: center\">You can&#8217;t move the exponent into a grouped sum because of the order of operations!!!!!<\/p>\n<p style=\"text-align: center\"><strong>INCORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}\\neq2^{2}+x^{2}[\/latex]<\/p>\n<p style=\"text-align: center\"><strong>\u00a0CORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}=\\left(2+x\\right)\\left(2+x\\right)[\/latex]<\/p>\n<\/div>\n<p>In the video that follows, you will see another examples of using a table to multiply two binomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Multiply Binomials Using a Table\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/tWsLJ_pn5mQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Further Examples<\/h2>\n<p>The next couple of examples show you some different forms binomials can take. \u00a0In the first, we will square a binomial that has a coefficient in front of the variable, like the product in the first example on this page. In the second we will find\u00a0the product of two binomials that have the variable on the right instead of the left.\u00a0We will use both\u00a0the FOIL method and the table method to simplify.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the product.\u00a0[latex]\\left(2x+6\\right)^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q255359\">Show Solution<\/span><\/p>\n<div id=\"q255359\" class=\"hidden-answer\" style=\"display: none\">We will use the FOIL method.<br \/>\n[latex]\\left(2x+6\\right)^{2}=\\left(2x+6\\right)\\left(2x+6\\right)[\/latex]<br \/>\n[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,2x\\left(2x\\right)=4x^{2}\\\\\\text{Outer}:\\,\\,\\,2x\\left(6\\right)=12x\\\\\\text{Inner}:\\,\\,\\,6\\left(2x\\right)=12x\\\\\\text{Last}:\\,\\,\\,\\,\\,6\\left(6\\right)=36\\end{array}[\/latex]<\/p>\n<p>Now you can collect the terms and simplify:<br \/>\n[latex]\\begin{array}{c}4x^2+12x+12x+36\\\\\\text{ }\\\\=4x^2+24x+36\\end{array}[\/latex]<\/p>\n<p>Answer<\/p>\n<p>[latex](2x+6)^{2}=4x^{2}+24x+36[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the last example, we want to show you another common form a binomial can take, each of the terms in the two binomials is the same, but the signs are different. You will see that in this case, the middle term will disappear.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply the binomials. [latex]\\left(x+8\\right)\\left(x\u20138\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q812247\">Show Solution<\/span><\/p>\n<div id=\"q812247\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(x\\right)=x^{2}\\\\\\text{Outer}:\\,\\,\\,\\,\\,\\,x\\left(-8\\right)=-8x\\\\\\text{Inner}:\\,\\,\\,\\,\\,\\,\\,8\\left(x\\right)=+8x\\\\\\text{Last}:\\,\\,\\,\\,\\,\\,\\,\\,\\,8\\left(-8\\right)=-64\\end{array}[\/latex]<\/p>\n<p>Add the terms. Note how the two x terms are opposites, so they sum to zero.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}x^{2}\\underbrace{-8x+8x}-64\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\text{ }\\\\=x^2-64\\\\\\text{ }\\\\\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(x+8\\right)\\left(x-8\\right)=x^{2}-64[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p style=\"text-align: left\">There are predictable outcomes when you square a binomial sum or difference. In general terms, for a binomial difference,<\/p>\n<p style=\"text-align: center\">[latex]\\left(a-b\\right)^{2}=\\left(a-b\\right)\\left(a-b\\right)[\/latex],<\/p>\n<p style=\"text-align: left\">the resulting product, after being simplified, will look like this:<\/p>\n<p style=\"text-align: center\">[latex]a^2-2ab+b^2[\/latex].<\/p>\n<p style=\"text-align: left\">The product of a binomial sum will have the following predictable outcome:<\/p>\n<p style=\"text-align: center\">[latex]\\left(a+b\\right)^{2}=\\left(a+b\\right)\\left(a+b\\right)=a^2+2ab+b^2[\/latex].<\/p>\n<p>The\u00a0product of a binomial sum and binomial difference of the same two monomial will have the following predictable outcome:<\/p>\n<p style=\"text-align: center\">[latex]\\left(a+b\\right)\\left(a-b\\right)=a^2-b^2[\/latex].<\/p>\n<p style=\"text-align: left\">Note that a and b in these generalizations could be integers, fractions, or variables with any kind of constant. \u00a0You will learn more about predictable patterns from products of binomials in later math classes.<\/p>\n<\/div>\n<p>In this section we showed how to multiply two binomials using the distributive property, an area model, by using a table, using the FOIL method, and the vertical method.\u00a0 Practice each method, and try to decide which one you prefer.<\/p>\n<p>Some of the forms a product of two binomials can take are listed here:<\/p>\n<ul>\n<li>[latex]\\left(x+5\\right)\\left(2x-3\\right)[\/latex]<\/li>\n<li>[latex]\\left(x+7\\right)^{2}[\/latex]<\/li>\n<li>[latex]\\left(x-1\\right)^{2}[\/latex]<\/li>\n<li>[latex]\\left(2-y\\right)\\left(5+y\\right)[\/latex]<\/li>\n<li>[latex]\\left(x+9\\right)\\left(x-9\\right)[\/latex]<\/li>\n<li>[latex]\\left(2x-4\\right)\\left(x+3\\right)[\/latex]<\/li>\n<\/ul>\n<p>And this is just a small list, the possible combinations are endless. For each of the products in the list, using one of the two methods presented here will work to simplify.<\/p>\n<h2>Summary<\/h2>\n<p>Multiplication of binomials and polynomials requires an understanding of the distributive property, rules for exponents, and a keen eye for collecting like terms. Whether the polynomials are monomials, binomials, or trinomials, carefully multiply each term in one polynomial by each term in the other polynomial. Be careful to watch the addition and subtraction signs and negative coefficients. A product is written in simplified form if all of its like terms have been combined.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-10837\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 146210, 146209,  146208, 146207. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Multiply Binomials Using An Area Model and Using Repeated Distribution. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/u4Hgl0BrUlo\">https:\/\/youtu.be\/u4Hgl0BrUlo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21046,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID 146210, 146209,  146208, 146207\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Multiply Binomials Using An Area Model and Using Repeated Distribution\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/u4Hgl0BrUlo\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"900491ffb15747d7abbbe31f06329d3c, 6eec43a415624829ab60fed521b80cd3","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-10837","chapter","type-chapter","status-publish","hentry"],"part":8336,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10837","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/users\/21046"}],"version-history":[{"count":33,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10837\/revisions"}],"predecessor-version":[{"id":20396,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10837\/revisions\/20396"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/8336"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10837\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/media?parent=10837"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=10837"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=10837"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/license?post=10837"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}